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IMA Journal of Numerical Analysis Advance Access originally published online on July 8, 2008
IMA Journal of Numerical Analysis 2009 29(3):746-759; doi:10.1093/imanum/drn036
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© The author 2008. Published by Oxford University Press on behalf of the Institute of Mathematics and its Applications. All rights reserved.

Local convergence of Newton's method in Banach space from the viewpoint of the majorant principle

Orizon P. Ferreira{dagger}

Instituto de Matemática e Estatística, Universidade Federal de Goiás, Campus II, Caixa Postal 131, CEP 74001-970, Goiânia, GO, Brazil

{dagger} Email: orizon{at}mat.ufg.br

Received on 25 July 2007. Revised on 26 April 2008.


  Abstract

A local convergence analysis of Newton's method for solving nonlinear equations, based on Kantorovich's majorant principle, is presented in this paper. This analysis provides a clear relationship between the majorant function, which relaxes the Lipschitz continuity of the derivative, and the nonlinear operator under consideration. It also allows us to obtain the optimal convergence radius, the biggest range for the uniqueness of the solution, and to unify some previous and unrelated results.

Key Words: Newton's method; majorant principle; local convergence; Banach space


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